Updated fluid equations

Below are the governing equations for the fluid:

.. math:: \frac{\partial (\epsilon_g \rho_g)}{\partial t} + \nabla \cdot (\epsilon_g \rho_g U_g)  = 0

Conservation of fluid mass:

.. math:: \frac{\partial (\varepsilon_g \rho_g)}{\partial t} + \nabla \cdot (\varepsilon_g \rho_g U_g)  = 0

  :label: eqn::mass_cons

.. math:: \frac{ \partial (\epsilon_g \rho_g U)}{\partial t} + \nabla \cdot (\epsilon_g \rho_g U_g U_g) + \epsilon_g \nabla p_g = \nabla \cdot \tau + {\bf g}

Conservation of fluid momentum:

.. math:: \frac{ \partial (\varepsilon_g \rho_g U)}{\partial t} + \nabla \cdot (\varepsilon_g \rho_g U_g U_g) + \varepsilon_g \nabla p_g = \nabla \cdot \tau + {\bf g} 

           + \sum_{part} \beta (V_{part} - U_g)

  :label: eqn::mom_cons

.. math:: \frac{\partial \epsilon_g}{\partial t} + \nabla \cdot (\epsilon_g  U_g)  = 0

  :label: eqn::mass_eps

Conservation of fluid volume:

where $\sum_{part} \beta (V_{part} - U_g)$ is the drag term in which $V_{part}$ represents the particle velocity.

.. math:: \frac{\partial \varepsilon_g}{\partial t} + \nabla \cdot (\varepsilon_g  U_g)  = 0

  :label: eqn::mass_vareps